A closed walk, with repeated vertices allowed. See path (graph theory). (This usage is common in computer science. In graph theory it is more often called a closed walk.)
A closed (simple) path, with no other repeated vertices than the starting and ending vertices. (This usage is common in graph theory, see "Cycle graph") This may also be called a simple cycle, circuit, circle, or polygon.
A closed directed walk, with repeated vertices allowed. (This usage is common in computer science. In graph theory it is more often called a closed directed walk.)
A closed directed (simple) path, with no repeated vertices other than the starting and ending vertices. (This usage is common in graph theory.) This may also be called a simple (directed) cycle.
The edge set of an undirected closed path without repeated vertices. This may also be called a circuit, circle, or polygon.
An element of the binary or integral (or real, complex, etc.) cycle space of a graph. (This is the usage closest to that in the rest of mathematics, in particular algebraic topology.) Such a cycle may be called a binary cycle, integral cycle, etc.
An edge set which has even degree at every vertex; also called an even edge set or, when taken together with its vertices, an even subgraph. This is equivalent to a binary cycle, since a binary cycle is the indicator function of an edge set of this type.
Chordless cycles are also sometimes called graph holes. A graph antihole is the complement of a graph hole.
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